I was puzzeling with the pseudosphere , a surface that except for a cusp has a constant negative Gaussian curvature, but has not everywhere the same mean curvature. (https://en.wikipedia.org/wiki/Pseudosphere )
This made me wonder are there also surfaces that have (for a largish area) a constant positive Gaussian curvature but not a constant mean curvature?
Something like a surface that has principle curvatures $K_1=1 ,K_2=1$ for a single point or limited set of curves while for the other points in this neightboorhood the Gaussian curvature $K_1K_2 =1$ but $ K_1 \not = K_2$ and this not just at a small part of the surface but at a largish area, like an area bounded by a cusp or boundary.
The surface does not need to be closed.
Theorem: If $M$ is a $C^{2}$ surface in $E^{3}$ having constant positive Gaussian curvature, and if $M$ is not part of a sphere, then the mean curvature of $M$ is non-constant.
Proof: Assume, contrapositively, that $M$ has constant Gaussian and mean curvatures. If $k_{1}$ and $k_{2}$ denote the principal curvature functions, then the sum and product $k_{1} + k_{2}$, $k_{1}k_{2}$ are both constant, so $$ (k_{1} - k_{2})^{2} = (k_{1} + k_{2})^{2} - 4k_{1}k_{2} $$ is constant, and therefore $k_{1} - k_{2}$ is constant. Since $k_{1} \pm k_{2}$ are constant, the principal curvatures themselves are constant.
Since $k_{1}k_{2} > 0$ by hypothesis, it follows that $k_{1} \equiv k_{2}$, i.e., $M$ is totally umbilic, and hence part of a sphere.